2016 AMC 12B Problem 7

Question

Josh writes the numbers

1,

2,

3,

\ldots,

99,

100.

He marks out

1,

skips the next number

(2),

marks out

3,

and continues skipping and marking out the next number to the end of his list. Then he goes back to the start of his list, marks out the first remaining number

(2),

skips the next number

(4),

marks out

6,

skips

8,

marks out

10,

and so on to the end. Josh continues in this manner until only one number remains. What is that number?

13

32

56

64

96

Answer: D

Concepts:power of 2 pattern recognition

Difficulty rating: 1440

Small hint:

After the first pass only the multiples of

2

remain; after the next, only the multiples of

4

Big hint:

Each pass keeps the multiples of the next power of

2;

find the highest power of

2

that is at most

100

Solution:

The first pass removes the odd numbers, leaving the multiples of

2.

The second pass removes

2,6,10,\ldots,

leaving the multiples of

4.

In general, after the

n

th pass only the multiples of

2^n

remain. The surviving number is the highest power of

2

not exceeding

100,

which is

2^6=64.

Thus, the correct answer is D.

2016 AMC 12B Problem 7

Answer: D

Difficulty rating: 1440

Small hint:

Big hint:

Solution:

Problem 7 in other years